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The structure of groups of prime power order. (English) Zbl 1008.20001

London Mathematical Society Monographs. New Series. 27. Oxford: Oxford University Press. xii, 334 p. (2002).
In his fundamental paper ‘A contribution to the theory of groups of prime power order’ [Proc. Lond. Math. Soc., II. Ser. 36, 29-95 (1933; Zbl 0007.29102)], Philip Hall writes: “The pages that follow represent the first stages of an attempt to construct a systematic general theory of groups of prime-power order. It is widely recognized, I believe, that the astonishing multiplicity and variety of these groups is one of the main difficulties which beset the advance of finite-group-theory.” Hall goes on by saying that because of this great variety people have been led to study special classes of such groups, and continues “With researches such as these, the present paper is not immediately concerned. I have tried, on the contrary, to discover some of the more obvious features that underlie the structure of the most general \(p\)-group.” (In this review a (finite) \(p\)-group is a group of order a power of the prime \(p\).) The book under review deals with the exciting developments the theory of groups of prime power order has experienced in the last 30 years. (Burnside groups are not dealt with.) The way we look at these groups has changed dramatically – as a result, a fundamental step in the program outlined by Philip Hall has been reached in that we can indeed claim to be understanding to some extent the general structure of \(p\)-groups. Norman Blackburn (a student of Philip Hall’s, and a crucial contributor to the theory) has been quoted saying that the proof of the coclass conjectures A-E (more about these later) represents “the first general theorem of the theory of \(p\)-groups.” The coclass of a finite \(p\)-group of order \(p^n\) and nilpotence class \(c\) is defined as \(n-c\). These conjectures were inspired by the work of the authors of the book under review on \(p\)-groups of maximal class. (See for instance [Q. J. Math., Oxf. II. Ser. 35, 293-304 (1984; Zbl 0547.20013)].) The authors credit M. F. Newman for the correct form of these conjectures; in particular, it is Newman who realized that space groups were playing a role here. Following work of Leedham-Green, McKay, W. Plesken, S. Donkin, A. Mann, A. Lubotzky, A. Shalev and E. I. Zelmanov, complete proofs of the coclass conjectures have been given by C. R. Leedham-Green [J. Lond. Math. Soc., II. Ser. 50, No. 1, 43-48 (1994; Zbl 0817.20031)] and A. Shalev [Invent. Math. 115, No. 2, 315-345 (1994; Zbl 0795.20009)].
In the Preface the authors begin by making clear that “[t]here is no hope of finding a finite set of invariants that will define every \(p\)-group up to isomorphism in a useful way, so we have to find a way around this difficulty.” This difficulty, which pops up already in the words of Hall quoted above, is apparent when one tries to count (pairwise non-isomorphic) \(p\)-groups of small order. For instance, there are \(56 092\) groups of order 256 [E. A. O’Brien, J. Algebra 143, No. 1, 219-235 (1991; Zbl 0734.20001)], and \(10 494 213\) of order 512 [B. Eick and E. A. O’Brien, J. Aust. Math. Soc., Ser. A 67, No. 2, 191-205 (1999; Zbl 0979.20021)]. As at least \(8 785 772\) of the latter groups have nilpotence class at most 2, any attempt at a classification by increasing order and class is doomed. One of the surprising points of this theory is that regularity is instead to be found among, roughly speaking, groups of high class – as it is the case if one fixes the coclass and lets the order grow.
Coming to the table of contents, the first two chapters deal with preliminaries and some group-theoretic constructions. The theory of \(p\)-groups of maximal class is presented in Chapter 3. This theory stems from a seminal paper of N. Blackburn [Acta Math. 100, 45-92 (1958; Zbl 0083.24802)]. Besides the essential work of the authors mentioned above, an important contribution has been provided by the unpublished 1971 Ph.D. thesis of R. Shepherd at the University of Chicago. Also, G. A. Fernández-Alcober et al. (see [J. Algebra 174, No. 2, 523-530 (1995; Zbl 0835.20030)]) have given sharp bounds for the so-called degree of commutativity of these groups. Uniserial actions follow in Chapter 4; they occur naturally in groups of maximal class, and more generally of finite coclass. The use of associated Lie algebras is introduced in Chapter 5. Powerful \(p\)-groups (see the book by J. D. Dixon, M. P. F. du Sautoy, A. Mann and D. Segal [Analytic pro-\(p\) groups. Revised and enlarged by Marcus du Sautoy and Dan Segal. 2nd ed. (Cambridge Studies in Advanced Mathematics. 61. Cambridge: Cambridge University Press) (1999; Zbl 0934.20001)]) are presented in Chapter 6, and used to give a proof of Conjecture A. This states that there is a function \(f\) of two variables such that every finite \(p\)-group of coclass \(r\) has a normal subgroup of class at most two and index at most \(f(p,r)\). Pro-\(p\)-groups (of finite coclass) are introduced in Chapter 7. Note that Conjectures C-E deal indeed with pro-\(p\)-groups (of finite coclass). They state that given \(p\) and \(r\), there are finitely many isomorphism classes of infinite pro-\(p\)-groups of coclass \(r\), and all of these groups are soluble. Also, these groups are (uniserial) \(p\)-adic pre-space groups. In Chapter 8 the construction methods of the authors for \(p\)-groups (of maximal class) are presented and extended. After the relevant material on homological algebra has been presented in Chapter 9, uniserial \(p\)-adic space groups are treated in Chapter 10. Chapter 11 aims at extending the theory of \(p\)-groups of maximal class to those of finite coclass. It is shown here that if \(P\) is a finite \(p\)-group of coclass \(r\), then \(P\) has a normal subgroup \(N\) of index bounded by a function of \(p\) and \(r\), such that \(P/N\) is constructible in the sense of Chapter 8. The last chapter discusses conditions weaker than being of finite coclass. Finite rank and obliquity are introduced; the Grigorchuk group and the Nottingham group are introduced and studied.
One does not expect a wealth of new material in a book, even in a research book of this level. However, there is considerable new material in this book, stuff that has been presented so far only in talks and lectures. Also, recent results from the literature are often given new proofs – the intent being always to make the material more understandable. This applies also to some material that has long found its way in textbooks. For instance at the beginning of the chapter on Homological Algebra, on page 171 the authors say “Spectral sequences are quite complicated; our treatment of them is original, and it is intended to be as elementary as possible, subject to obtaining the results we need.”
Which brings us to the style of the book, which we like very much. The book has to get technical at times – sometimes very much so. However, the authors are very careful not to add unnecessary complications, in particular, as it is clear from the previous quotation, by resisting the temptation of going encyclopedic. For instance, the proof of Theorem 1.1.30 on the expansion of powers in a free group begins with “Since we do not wish to distract the reader from the main thrust of this book, we shall not prove the uniqueness of the \(m_i\), but only that \((xy)^n\) can be written in this form. This will suffice for our purpose.” Another instance comes on page 172: “We do not consider products in cohomology. They play an important role in the more advanced theory of spectral sequences. There is nothing particularly hard about products, but we do not need them.” At this point we might perhaps quote the closing sentence of the book, which sounds “We assure […] that this book contains no jokes, intentional or otherwise.” In the opinion of this reviewer, it is wise to take this statement at full face value (well, not quite). For instance, on page 214 the beautiful \(p\)-group generation algorithm of M. F. Newman [Group Theory, Canberra 1975, Lect. Notes Math. 573, 73-84 (1977; Zbl 0519.20018)] and E. A. O’Brien [J. Symb. Comput. 9, No. 5/6, 677-698 (1990; Zbl 0736.20001)] is described; this algorithm, which makes subtle use of cohomology, has led to the spectacular counting results for \(p\)-groups of low order mentioned above. The text goes on by saying “This method is used for constructing \(p\)-groups by computer […]. Unfortunately (or fortunately) the cohomological machine is not in general selective enough for this approach to be of much theoretical use. The method is too precise; it forces one to produce an exact analysis, omitting no detail. But the precise structure of \(p\)-groups is too complex for the human intellect; the art of studying \(p\)-groups lies in looking for general features that transcend minutiae.” This may elicit a smile at a cursory reading; there is nothing wrong with this, provided one does not miss that an important point is being made in a clear and concise way – a welcome occurrence in mathematical writing.
It is very difficult for a book to be error-free. In the case of the book under review, we believe the authors to be in error on page 314. They are considering the positive part of a twisted loop algebra \(L\) of the Witt algebra over the field with \(p\) elements, which has a certain basis \(f_i\), for \(i\geq 1\). They state that \(\text{ad}(f_i)^p=0\) for all \(x\in L\). Indeed, this is not true when \(p\) divides \(i\). If \(x=f_i\) (and \(\text{ad}(x)^p=0\)), they then write \(i_x =\exp(\text{ad}(x))=1+\text{ad}(x)+\cdots+\text{ad}(x)^{p-1}/(p-1)!\) and claim that this is an automorphism of \(L\). This, however, is not true in general, as it is clear if one works out the case \(p=2\), and in fact it can be seen not to hold for any such \(x\) here. (I am grateful to Sandro Mattarei for an enlightening discussion of this point. Note that the exponential of a derivation \(D\) is indeed an automorphism, in odd characteristic \(p\), if \(D^{(p+1)/2}=0\).) However, the authors do not pursue this point further, so no damage follows from this impossibility.
The book is beautifully produced – a pleasure to the eyes, as well as to the mind, to read. We found only one very minor typographical glitch, in that the spacing to the right of the symbol “\(\triangleleft\)” is sometimes not quite correct. For instance the occurrence in item (ii) of the statement of Lemma 4.1.13 has the wrong spacing, while the one in item (i) has the right one.
Reviewer: A.Caranti (Povo)

MSC:

20-02 Research exposition (monographs, survey articles) pertaining to group theory
20D15 Finite nilpotent groups, \(p\)-groups
20E18 Limits, profinite groups
20F12 Commutator calculus
20F40 Associated Lie structures for groups
20J06 Cohomology of groups
20F05 Generators, relations, and presentations of groups
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